Euclid’s Proof of the Infinitude of Primes: Distorted, Clarified, Made Obsolete, and Confirmed in Modern Mathematics
REINHARD SIEGMUND-SCHULTZE
n this article1 I reflect on the recurrent theme of Michael Hardy and Catherine Woodgold (henceforth modernizing historical mathematical proofs, vocabulary, H&W) entitled ‘‘Prime Simplicity’’ appeared in this journal II and symbolism, and the extent to which this modern- in 2009. H&W discuss Euclid’s proof mainly from the ization serves to clarify, is able to preserve, or is bound to point of view of its distortion by modern authors who distort the original meaning. My example is also a quite often claim that it had the form of a reductio ad recurrent one: Euclid’s proof of the infinitude of primes absurdum.3 in book IX, theorem 20 of his Elements. Elementary My article, on the other hand, aims at a systematic pre- number theory is a very appropriate field for discussing sentation and logical analysis of Euclid’s proof (as it is such general historiographic questions on a nontechnical preserved in the critical editions by Euclid scholars) and, level.2 Indeed, a widely quoted and stimulating article by above all, at a more detailed discussion of its interpretation
1I dedicate this article to Walter Purkert (Bonn), the coordinator and very active force behind the excellent Felix Hausdorff edition soon to be completed, on the occasion of his 70th birthday in 2014. 2Mesˇ trovic´ (2012) contains many details on modern proofs of the infinitude of primes but is less interested in Euclid’s original theorem and its modern variants. Most of his material goes far beyond the simple fact of the infinitude of primes that is the focus of the present paper. 3In the following, the notions reductio ad absurdum, proof by contradiction, and indirect proof are used interchangeably for assuming the logical opposite of what one wants to prove and then reaching, by legitimate logical steps, a contradiction that requires the assumption of the original claim.